Problem 27 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{6 x+6}{3 x-3} $$

Step-by-Step Solution

Answer

Simplified form: \[ \frac{2(x + 1)}{x - 1} \]

Step 1: Factor out the Greatest Common Factor (GCF)

First, identify the GCF in the numerator and the denominator. In the numerator, both terms have a GCF of 6, and in the denominator, both terms have a GCF of 3. Therefore, factor out these GCFs. \[ \frac{6x + 6}{3x - 3} = \frac{6(x + 1)}{3(x - 1)} \]

Step 2: Simplify the Fraction

Next, simplify the fraction by dividing both the numerator and the denominator by the common factors. The fraction \( \frac{6}{3} \) reduces to 2.\[ \frac{6(x + 1)}{3(x - 1)} = \frac{2 \times 3(x + 1)}{1 \times 3(x - 1)} = \frac{2(x + 1)}{(x - 1)} \]

Step 3: Write the Final Simplified Form

Since there are no more common factors to cancel out, the expression is already simplified. The final simplified form is: \[ \frac{2(x + 1)}{x - 1} \]

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest number that divides exactly into two or more numbers. To simplify algebraic fractions, identifying the GCF is crucial. For example, in the expression \( \frac{6x + 6}{3x - 3} \), the GCF of 6 and 6 in the numerator is 6. Similarly, the GCF of 3 and -3 in the denominator is 3. By factoring out the GCF, we can rewrite the expression as \( \frac{6(x + 1)}{3(x - 1)} \). Finding the GCF helps us to simplify the fractions more efficiently.

Factoring

Factoring is breaking down an expression into simpler components that, when multiplied together, give the original expression. In the context of simplifying algebraic fractions, factoring is essential. In the expression \( \frac{6(x + 1)}{3(x - 1)} \), we factored 6 from the numerator and 3 from the denominator. Factoring expressions can reveal common factors in the numerator and the denominator, which can then be simplified, making the overall fraction more manageable.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This involves dividing both the numerator and the denominator by their greatest common factor (GCF). In our example, \( \frac{6(x + 1)}{3(x - 1)} \) simplifies further because \( \frac{6}{3} = 2 \). Therefore, we get \( \frac{2(x + 1)}{x - 1} \). Simplifying fractions helps us work with more manageable numbers and makes further operations easier.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving at least one variable and sometimes numbers and operation symbols. For example, \( 6x + 6 \) and \( 3x - 3 \) are algebraic expressions. When simplifying algebraic fractions, understanding each part of the expression and how to factor them is vital. Factoring transforms the expressions into a simpler form and makes it easier to identify and cancel out common factors.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are algebraic expressions. Simplifying rational expressions often involves factoring both the numerator and the denominator to find common factors. In our example, \( \frac{6x + 6}{3x - 3} \) is a rational expression. By factoring and simplifying, we reduce it to \( \frac{2(x + 1)}{x - 1} \), which is its simplest form. Simplifying rational expressions is crucial for solving equations and understanding higher-level algebra concepts.